= Orthonormal frame bundle
{title2=$O(M,g)$}
= Pseudo-orthonormal frame bundle
{synonym}
For a metric of signature $(r,s)$, the orthonormal frame bundle consists of all linear isometries $u:\mathbb R^{r,s}\to T_xM$. Right composition by $O(r,s)$ makes it a <principal bundle>. Its standard <associated vector bundle> is the <tangent bundle>, via $[u,v]\mapsto u(v)$. A global orthonormal frame is a global section and therefore trivializes this principal bundle. A <Lie group> with a left-invariant metric has such a frame by left translating an orthonormal basis at the identity.
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